MinT - Best Known (54−17, 54, s)-Nets in Base 7

Best Known (54−17, 54, s)-Nets in Base 7

(54−17, 54, 204)-Net over F₇ — Constructive and digital

Digital (37, 54, 204)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 18, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 9, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]
2. digital (19, 36, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 18, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(54−17, 54, 814)-Net over F₇ — Digital

Digital (37, 54, 814)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−17, 54, 249060)-Net in Base 7 — Upper bound on s

There is no (37, 54, 249061)-net in base 7, because

1 times m-reduction [i] would yield (37, 53, 249061)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 616 882886 976246 197666 672351 382292 005535 992577 > 7⁵³ [i]